Eardley  followed the first route and discussed the effects of a time variation of the gravitational constant on binary pulsar in the framework of the Brans–Dicke theory. In that case, both a dipole gravitational radiation and the variation of induce a periodic variation in the pulse period. Nordtvedt  showed that the orbital period changes as
Damour et al.  used the timing data of the binary pulsar PSR 1913+16. They implemented the effect of the time variation of by considering the effect on . They defined, in a phenomenological way, that , where is the part of the orbital period derivative that is not explained otherwise (by gravitational waves radiation damping). This theory-independent definition has to be contrasted with the theory-dependent result (138) by Nordtvedt . They got then reexamined the data of PSR 1913+16 and established the upper bound  used data from PSR B1913+16 and PSR B1855+09 respectively to get
All the previous results concern binary pulsars but isolated ones can also be used. Heintzmann and Hillebrandt  related the spin-down of the pulsar JP1953 to a time variation of . The spin-down is a combined effect of electromagnetic losses, emission of gravitational waves, possible spin-up due to matter accretion. Assuming that the angular momentum is conserved so that = constant, one deduces that then modelled the pulsar by a polytropic ) white dwarf and deduced that so that . Mansfield  assumed a relativistic degenerate, zero temperature polytropic star and got that, when , at a level. He also noted that a positive induces a spin-up counteracting the electromagnetic spin-down, which can provide another bound if an independent estimate of the pulsar magnetic field can be obtained. Goldman , following Eardley , used the scaling relations and to deduce that . He used the data from the pulsar PSR 0655+64 to deduce that the rate of decrease of was smaller than  of 10 years high precision timing data on the millisecond pulsar PSR J0437-4715 has allowed to improve the constraint to
Recently, it was argued [266, 432] that a variation of would induce a departure of the neutron star matter from -equilibrium, due to the changing hydrostatic equilibrium. This would force non-equilibrium -processes to occur, which release energy that is invested partly in neutrino emission and partly in heating the stellar interior. Eventually, the star arrives at a stationary state in which the temperature remains nearly constant, as the forcing through the change of is balanced by the ongoing reactions. Comparing the surface temperature of the nearest millisecond pulsar, PSR J0437-4715, inferred from ultraviolet observations, two upper limits for variation were obtained, , direct Urca reactions operating in the neutron star core are allowed, and , considering only modified Urca reactions. This was extended in  in order to take into account the correlation between the surface temperatures and the radii of some old neutron stars to get .
Living Rev. Relativity 14, (2011), 2
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